A note on k-potence preserving maps

Let M_n be the algebra of all n×n complex matrices and Γ_n the set of all k-potent matrices in M_n. Suppose ?:M_n→M_n is a map satisfying A-λB∈Γ_n implies ?(A)-λ?(B)∈Γ_n, where A, B∈M_n, λ∈C. Then either ? is of the form ?(A)=cTAT^-1, A∈M_n, or ? is of the form ?(A)=cTA^tT^-1, A∈M_n, where T∈M_n is an invertible matrix, c∈C satisfies c^k=c.     

Maximum A-isometric part of an A-contraction and applications

For two bounded linear operators A and T on a complex Hilbert space H (A being positive) which satisfy the inequality T*AT ≤ A, we study the maximum subspace ℳ₀ which reduces A and T, on which the equality T*AT = A holds. We show that in some cases involving the condition AT = A ^1/2 TA ^1/2, ℳ₀ can be expressed in terms of the minimal isometric dilation of the contraction $ \hat T $ \hat T on $ \overline {R(A)} $ \overline {R(A)} associated to T by the condition $ \hat T $ \hat T A ^1/2 = A ^1/2 T. As application we find a concrete representation for ℳ₀ when T is a contraction with S _T = S _T ², where S _T is the strong limit of the sequence [T ^*n T ⁿ : n ≥ 1]. Also, we derive some applications for hyponormal contractions and quasi-isometries.     

A structural characterization of real k-potent matrices

Some well-known characterizations of nonnegative k-potent matrices have been obtained by Flor [P. Flor, On groups of nonnegative matrices, Compositio Math. 21 (1969), pp. 376–382.] and Jeter and Pye [M. Jeter and W. Pye, Nonnegative (s,?t)-potent matrices, Linear Algebra Appl. 45 (1982), pp. 109–121.]. In this article, we obtain a structural characterization of a real k-potent matrix A, provided that (sgn(A)) ^k+1 is unambiguously defined, regardless of whether A is nonnegative or not.     

On the sequence of powers of a stochastic matrix with large exponent

We consider the class of primitive stochastic n×n matrices A, whose exponent is at least (n²−2n+2)/2+2. It is known that for such an A, the associated directed graph has cycles of just two different lengths, say k and j with k>j, and that there is an α between 0 and 1 such that the characteristic polynomial of A is λⁿ−αλ^n−j−(1−α)λ^n−k. In this paper, we prove that for any mn, if α1/2, then A^m+k−A^m_∞A^m−1w^T_∞, where 1 is the all-ones vector and w^T is the left-Perron vector for A, normalized so that w^T1=1. We also prove that if jn/2, n31 and , then A^m+j−A^m_∞A^m−1w^T_∞ for all sufficiently large m. Both of these results lead to lower bounds on the rate of convergence of the sequence A^m.     

On isometries of matrix spaces

Let A _kbe the group of isometries of the space of n-by-n matrices over reals (resp. complexes, quaternions) with respect to the Ky Fan k-norm (see the Introduction for the definitions). Let Γ⁰ be the group of transformations of this space consisting of all products of left and right multiplications by the elements of SO(n)(resp. U(n), Sp(n)). It is shown that, except for three particular casesA_k coincides with the normalizer of Γ in Δ group of isometries of the above matrix space with respect to the standard inner product. We also give an alternative treatment of the case D = R n = 4k = 2 which was studied in detail by Johnson, Laffey, and Li [4].     

Minkowski spaces with extremal distance from the Euclidean space

It is proved that if the Banach-Mazur distance between ann-dimensional Minkowski spaceB andl ₂ ⁿ satisfiesd (B ₁ l ₂ ⁿ ) ≧c √n (for some constantc>0 and for bign) thenB contains anA(c)-isomorphic copy ofl ₁ ^k (fork ∼ log log logn). In the special cased (B ₁ l ₂ ⁿ ) = √n,B contains an isometric copy ofl ₁ ^k fork ∼ logn.     

On the Proximal Point Algorithm

Let A be a maximal monotone operator in a real Hilbert space H and let {u _n } be the sequence in H given by the proximal point algorithm, defined by u _n =(I+c _n A)⁻¹(u _n−1−f _n ), ∀ n≥1, with u ₀=z, where c _n >0 and f _n ∈H. We show, among other things, that under suitable conditions, u _n converges weakly or strongly to a zero of A if and only if lim inf  _n→+∞|w _n |<+∞, where w _n =(∑ _k=1 ⁿ c _k )⁻¹∑ _k=1 ⁿ c _k u _k . Our results extend previous results by several authors who obtained similar results by assuming A ⁻¹(0)≠φ.     

On tree census and the giant component in sparse random graphs

For each of the two models of a sparse random graph on n vertices, G(n, # of edges = cn/2) and G(n, Prob (edge) = c/n) define t_n(k) as the total number of tree components of size k (1 ≤ k ≤ n). the random sequence {[t_n(k) - nh(k)]n^?1/2} is shown to be Gaussian in the limit n →∞, with h(k) = k^k?2c^k?1e^?kc/k! and covariance function being dependent upon the model. This general result implies, in particular, that, for c> 1, the size of the giant component is asymptotically Gaussian, with mean nθ(c) and variance n(1 ? T)^?2(1 ? 2Tθ)θ(1 ? θ) for the first model and n(1 ? T)^?2θ(1 ? θ) for the second model. Here Te^?T = ce^?c, T<1, and θ = 1 ? T/c. A close technique allows us to prove that, for c < 1, the independence number of G(n, p = c/n) is asymptotically Gaussian with mean nc^?1(β + β²/2) and variance n[c^?1(β + β²/2) ?c^?2(c + 1)β²], where βe^β = c. It is also proven that almost surely the giant component consists of a giant two-connected core of size about n(1 ? T)β and a “mantle” of trees, and possibly few small unicyclic graphs, each sprouting from its own vertex of the core.     

Abstract Characteristic Function

The characteristic function for a contraction is a classical complete unitary invariant devised by Sz.-Nagy and Foias. Just as a contraction is related to the Szego kernel k_S(z,w) = (1 - z [`(w)])^-1{k_S(z,w) = (1 - z {\overline {w}})^{-1}} for |z|, |w| < 1, by means of (1/k _S )(T, T*) ≥ 0, we consider an arbitrary open connected domain Ω in \mathbb Cⁿ{{\mathbb {C}}^n}, a kernel k on Ω so that 1/k is a polynomial and a tuple T = (T ₁, T ₂, . . . , T _n ) of commuting bounded operators on a complex separable Hilbert space H{\mathcal H} such that (1/k)(T, T*) ≥ 0. Under some standard assumptions on k, it turns out that whether a characteristic function can be associated with T or not depends not only on T, but also on the kernel k. We give a necessary and sufficient condition. When this condition is satisfied, a functional model can be constructed. Moreover, the characteristic function then is a complete unitary invariant for a suitable class of tuples T.     

Tensor Product Representations of the Lie Superalgebra 𝔭(n) and Their Centralizers

Abstract

We construct an associative algebra A _k and show that there is a representation of A _k on V ^?k , where V is the natural 2n-dimensional representation of the Lie superalgebra 𝔭(n). We prove that A _k is the full centralizer of 𝔭(n) on V ^?k , thereby obtaining a “Schur-Weyl duality” for the Lie superalgebra 𝔭(n). This result is used to understand the representation theory of the Lie superalgebra 𝔭(n). In particular, using A _k we decompose the tensor space V ^?k , for k = 2 or 3, and show that V ^?k is not completely reducible for any k ≥ 2.     

The asymptotic number of labeled connected graphs with a given number of vertices and edges

Let c(n, q) be the number of connected labeled graphs with n vertices and q ≤ N = (2n ) edges. Let x = q/n and k = q ? n. We determine functions w_k ? 1. a(x) and φ(x) such that c(n, q) ? w_k(q^N)eⁿφ(x)+a(x) uniformly for all n and q ≥ n. If ? > 0 is fixed, n→ ∞ and 4q > (1 + ?)n log n, this formula simplifies to c(n, q) ? (N^q) exp(–ne^?2q/n). on the other hand, if k = o(n^1/2), this formula simplifies to c(n, n + k) ? 1/2 w_k (3/π)^1/2 (e/12k)^k/2nⁿ?^(3k?1)/2.     

Approximation Numbers of Operators on Normed Linear Spaces

In [1], B?ttcher et. al. showed that if T is a bounded linear operator on a separable Hilbert space H, {e_j}_j=1^￥H, \{e_{j}\}_{j=1}^{\infty} is an orthonormal basis of H and P_n is the orthogonal projection onto the span of {e_j}_j=1ⁿ\{e_{j}\}_{j=1}^{n}, then for each k ? \mathbbNk \in {\mathbb{N}}, the sequence {s_k(P_nTP_n)}\{s_{k}(P_{n}TP_{n})\} converges to s_k(T), where for a bounded operator A on H, s_k(A) denotes the kth approximation number of A, that is, s_k(A) is the distance from A to the set of all bounded linear operators of rank at most k − 1. In this paper we extend the above result to more general cases. In particular, we prove that if T is a bounded linear operator from a separable normed linear space X to a reflexive Banach space Y and if {P_n} and {Q_n} are sequences of bounded linear operators on X and Y, respectively, such that ||P_n|| ||Q_n|| ￡ 1\|P_n\| \|Q_n\| \leq 1 for all n ? \mathbbNn \in {\mathbb{N}} and {Q_nTP_n} converges to T under the weak operator topology, then {s_k(Q_nTP_n)}\{s_{k}(Q_{n}TP_{n})\} converges to s_k(T). We also obtain a similar result for the case of any normed linear space Y which is the dual of some separable normed linear space. For compact operators, we give this convergence of s_k(Q_nTP_n)s_{k}(Q_{n}TP_{n}) to s_k(T) with separability assumptions on X and the dual of Y. Counter examples are given to show that the results do not hold if additional assumptions on the space Y are removed. Under separability assumptions on X and Y, we also show that if there exist sequences of bounded linear operators {P_n} and {Q_n} on X and Y respectively such that (i) Q_nTP_nQ_{n}TP_{n} is compact, (ii) ||P_n|| ||Q_n|| ￡ 1\|P_{n}\| \|Q_{n}\| \leq 1 and (iii) {Q_nTP_n}\{Q_{n}TP_{n}\} converges to T in the weak operator topology, then {s_k(Q_nTP_n)}\{s_k(Q_{n}TP_{n})\} converges to s_k(T) if and only if s_k(T) = s_k(T￠)s_{k}(T) = s_{k}(T^\prime). This leads to a generalization of a result of Hutton [3], proved for compact operators between normed linear spaces.     

Iterative Methods for Fixed Points of Asymptotically Weakly Contractive Maps

Suppose K is a closed convex nonexpansive retract of a real uniformly smooth Banach space E with P as the nonexpansive retraction. Suppose T : K → E is an asymptotically d-weakly contractive map with sequence {k_n }, k_n ≥ 1, lim k_n = 1 and with F(T) _n int (K) ≠ ø F(T):= {x ∈ K: Tx = x}. Suppose {x _n } is iteratively defined by x _n+1 = P((l ? k_nα_n )x _n +k _n α _n T(PT) ^n?l x_n ), n = 1,2,...,x ₁ ∈ K, where α_n∈ (0,l) satisfies lim α_n = 0 and Σα_n = ∞. It is proved that {x _n } converges strongly to some x ^* ∈ F(T)∩ int K. Furthermore, if K is a closed convex subset of an arbitrary real Banach space and T is, in addition uniformly continuous, with F(T) ≠ ø, it is proved that {x_n } converges strongly to some x ^* ∈ F(T).     

On the divisor function of sets with even partition functions

Summary For P∈ F₂[z] with P(0)=1 and deg(P)≧ 1, let A =A(P) be the unique subset of N (cf. [9]) such that Σ_n≧0 p(A,n)zⁿ ≡ P(z) mod 2, where p(A,n) is the number of partitions of n with parts in A. To determine the elements of the set A, it is important to consider the sequence σ(A,n) = Σ _{d|n, d∈A} d, namely, the periodicity of the sequences (σ(A,2^kn) mod 2^k+1)_n≧1 for all k ≧ 0 which was proved in [3]. In this paper, the values of such sequences will be given in terms of orbits. Moreover, a formula to σ(A,2^kn) mod 2^k+1 will be established, from which it will be shown that the weight σ(A₁,2^kz_i) mod 2^k+1 on the orbit <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>z_i$ is moved on some other orbit z_j when A₁ is replaced by A₂ with A₁= A(P₁) and A₂= A(P₂) P₁ and P₂ being irreducible in F₂[z] of the same odd order.     

Affine \mathbb{T} -varieties of complexity one and locally nilpotent derivations

Let X = Spec A be a normal affine variety over an algebraically closed field k of characteristic 0 endowed with an effective action of a torus \mathbbT \mathbb{T} of dimension n. Let also ∂ be a homogeneous locally nilpotent derivation on the normal affine \mathbbZⁿ {\mathbb{Z}^n} -graded domain A, so that ∂ generates a k ₊-action on X that is normalized by the \mathbbT \mathbb{T} -action.     

Monotonicity of permanents of direct sums of doubly stochastic matrices

Let Ωn be the set of all n × n doubly stochastic matrices, let J_n be the n × n matrix all of whose entries are 1/n and let σ _k (A) denote the sum of the permanent of all k × k submatrices of A. It has been conjectured that if A ε Ω _n and A ≠ J_J then g_A,k (θ) ? σ _k ((1 θ)J_n 1 θA) is strictly increasing on [0,1] for k = 2,3,…,n. We show that if A = A ₁ ⊕ ⊕A_t (t ≥ 2) is an n × n matrix where A_i for i = 1,2, …,t, and if for each i g_Ai,ki (θ) is non-decreasing on [0.1] for k_t = 2,3,…,n_i , then g_A,k (θ) is strictly increasing on [0,1] for k = 2,3,…,n.     

Graph decompositions without isolated vertices III

Let G be a graph of order n, and n = Σ^k_i=1 a_i be a partition of n with a_i ≥ 2. In this article we show that if the minimum degree of G is at least 3k−2, then for any distinct k vertices v₁,…, v_k of G, the vertex set V(G) can be decomposed into k disjoint subsets A₁,…, A_k so that |A_i| = a_i,v_iisA_i is an element of A_i and “the subgraph induced by A_i contains no isolated vertices” for all i, 1 ≥ i ≥ k. Here, the bound on the minimum degree is sharp. © 1997 John Wiley & Sons, Inc.     

On the normal bundle of curves on nodal quartic surfaces

Let k be an algebraically closed field, char k = 0. Let C be an irreducible nonsingular curve such that 2C = S ? F, where S and F are two surfaces and all the singularities of F are rational double points (if any). We prove that C can never pass through rational singularities of types A _2n n∈N, E₆ and E₈. We give conditions for C to pass through rational singularities of types. A _2k+1 k∈Z⁺ D_n n≥4 and E₇, (0.8).     

Minimal cogenerators need not be unique

A_n = A_n(?) denotes the unique left distributive binary system on {0, 1,…,2ⁿ?1) that satisfies a ? 1 = a + 1 mod 2ⁿfor all a ? A_n, and o_n(a) = k indicates the period 2^kof a ? A_n (if b,c ? A_n, then a ? b = a ? c iff b ‵ c mod 2^kand if 0 < b < c < 2^kthen a < a ? b < a ? c). Amongs others, we prove that o_t2(2^t?2) ≤ t holds for every integer t ≥ 2, the equality taking place iff t is of the form 2^2? for an integer s ≥ 0.     

Families of Finite Subsets of ℕ of Low Complexity and Tsirelson Type Spaces

We study Tsirelson type spaces of the form T[(ℳ︁_k, θ_k)^l_k=1] defined by a finite sequence (ℳ︁_k)^l_k=1 of compact families of finite subsets of ℕ. Using an appropriate index, denoted by i(ℳ︁), to measure the complexity of a family ℳ︁, we prove the following: If i(ℳ︁_k) < ω for all k = 1, …, l, then the space T[(ℳ︁_k, θ_k)^l_k=1] contains isomorphically some l_p, 1 < p < ∞, or c₀. If i(ℳ︁) = ω, then the space T[ℳ︁, θ] contains a subspace isomorphic to a subspace of the original Tsirelson's space.